Revisiting Clifford Algebras

Abstract: This paper is divided in three parts. In the first part, I study the Clifford algebra associated to the hessian of a functional $f$ defined on an open subset of $\mathbb{R}^n$ \ and the Clifford algebra associated to the hessian of the Legendre transform of $f$. I give also the definition of a tensorial topology on a Clifford algebra. In the second part, I study the Clifford algebra $C\left( E,q\right) $ of an infinite dimensional Banach space $E$ and its main properties. Finally, in the third part, I give the explicit formula of the hilbertian kernel of a Clifford algebra with examples.